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Physica Scripta

Perfect state transfer by means of discrete-time quantum walk on the complete bipartite graph

J.S.T. Huang, Dan Li, Panlong Li, Yuqian Zhou, Yu‐Guang Yang

Published Dec 7, 2023• 1 citation
Publisher

Abstract

Abstract Perfect state transfer has attracted a great deal of attention recently due to its crucial role in quantum communication and scalable quantum computation. In this paper, we propose the perfect state transfer algorithms with a pair of sender-receiver and two pairs of sender-receiver on the complete bipartite graph respectively. The algorithm with a pair of sender-receiver is implemented through discrete-time quantum walk, flexibly setting the coin operators based on the positions of the sender and receiver. The algorithm with two pairs of sender-receiver ensures that the two quantum states are distributed on both sides of the complete bipartite graph during the process, thereby achieving perfect state transfer. In addition, the quantum circuits corresponding to the algorithms are provided. The algorithms can transfer an arbitrary quantum state and can simultaneously transfer two arbitrary quantum states from the senders to the receivers in any case. Moreover, the algorithms are not only applicable to complete bipartite graphs but also to more graph structures with complete bipartite subgraphs, which will provide potential applications for quantum information processing.

DOI
10.1088/1402-4896/ad137a

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Abstract Perfect state transfer has attracted a great deal of attention recently due to its crucial role in quantum communication and scalable quantum computation. In this paper, we propose the perfect state transfer algorithms with a pair of sender-receiver and two pairs of sender-receiver on the complete bipartite graph respectively. The algorithm with a pair of sender-receiver is implemented through discrete-time quantum walk, flexibly setting the coin operators based on the positions of the sender and receiver. The algorithm with two pairs of sender-receiver ensures that the two quantum states are distributed on both sides of the complete bipartite graph during the process, thereby achieving perfect state transfer. In addition, the quantum circuits corresponding to the algorithms are provided. The algorithms can transfer an arbitrary quantum state and can simultaneously transfer two arbitrary quantum states from the senders to the receivers in any case. Moreover, the algorithms are not only applicable to complete bipartite graphs but also to more graph structures with complete bipartite subgraphs, which will provide potential applications for quantum information processing.
Abstract Previously it was shown that (almost) perfect state transfer can be achieved on the complete bipartite graph by a discrete-time coined quantum walk based algorithm when both the sender and receiver vertices are in the same partition of the graph and when the sender and receiver are in opposite partitions of the same size. By changing the coin operator, we analyze the state transfer problem and we show that it is still possible to achieve state transfer with high fidelity even when the sender and receiver are in different partitions with different sizes. Moreover, it is also possible to use an active switch approach using lackadaisical quantum walks where the marked vertex is switched between the sender and receiver during the algorithm.
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Perfect state transfer between two marked vertices of a graph by means of a discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that, for the coined quantum walks on a star graph and a complete graph with self-loops, perfect state transfer between the sender and receiver vertex is achieved for an arbitrary number of vertices NNN in O(N)O(\sqrt{N})O(N​) steps of the walk. Finally, we show that Szegedy's walk with queries on a complete graph allows for state transfer with unit fidelity in the limit of large NNN.
We show that a perfect state transfer of an arbitrary unknown two-qubit state can be achieved via a discrete-time quantum walk with various settings of coin flips and extend this method to the distribution of an arbitrary unknown multiqubit entangled state between every pair of sites in the multidimensional network. Furthermore, we study the routing of quantum information on this network in a quantum-walk architecture, which can be used as quantum information processors to communicate between separated qubits.
Abstract Continuous-time quantum walks (CTQW) becoming more significant for quantum information and quantum computation. The perfect state transfer (PST) on graphs is a specific phenomena of CTQW which is widely used in quantum communication of small system, and be found critical for quantum algorithm design recently, such as quantum approximate optimization algorithms and quantum search. If a network GGG contains 2N2N2N vertices, and PST occurs between NNN pairs of vertices at the same time TTT, then we say GGG admits symmetric PST (SPST). In this work, we will study the SPST on bipartite graphs via spectra. And then, we will present how to set couplings of special bipartite graphs such as cycles, grids and chains to admit SPST is given. For cycles with 6, 8, 10, 12 vertices, we give the analytical method to obtain the couplings to admit SPST. Analogously, the couplings for chains and grids to admit SPST are analytical obtained. We also present a typical approach by using the graph tensor product to construct new quantum networks that admit SPST, in which the new quantum networks are derived from special kinds of integral graphs, such as complete graphs, strongly regular graphs and distance regular graphs.
Search and state transfer between hubs, i.e., fully connected vertices, on otherwise arbitrary connected graph is investigated. Motivated by a recent result of Razzoli et al. [J. Phys. A: Math. Theor. 55, 265303 (2022)] on the universality of hubs in continuous-time quantum walks and spatial search, we extend the investigation to state transfer and also to the discrete-time case. We show that the continuous-time quantum walk allows for perfect state transfer between multiple hubs if the numbers of senders and receivers are close. Turning to the discrete-time case, we show that the search for hubs is successful provided that the initial state is locally modified to account for a degree of each individual vertex. Concerning state transfer using discrete-time quantum walk, it is shown that between a single sender and a single receiver one can transfer two orthogonal states in the same run time. Hence, it is possible to transfer an arbitrary quantum state of a qubit between two hubs. In addition, if the sender and the receiver know each other location, another linearly independent state can be transferred, allowing for exchange of a qutrit state. Finally, we consider the case of transfer between multiple senders and receivers. In this case we cannot transfer specific quantum states. Nevertheless, quantum walker can be transferred with high probability in two regimes---either when there is a similar number of senders and receivers, which is the same as for the continuous-time quantum walk, or when the number of receivers is considerably larger than the number of senders. Our investigation is based on dimensional reduction utilizing the invariant subspaces of the respective evolutions and the fact that for the appropriate choice of the loop weights the problem can be reduced to the complete graph with loops.